Eevee film grain type filter?
There's currently no way to emulate film grain without the compositor.

This is obviously a crude workaround, but you could find a film grain image sequence or video, (or somehow make one procedurally using nodes), and use it as an alpha texture for a plane. Then, constrain the plane's rotation and location to always be right in front of the camera. This wouldn't work if you need Depth of Field, but it would probably work ok if you didn't use DoF.